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Decomposition of positive sesquilinear forms and the central decomposition of gauge-invariant quasi-free states on the Weyl-algebra. (English) Zbl 0807.46088
Summary: A decomposition theory for positive sesquilinear forms densely defined in Hilbert spaces is developed. On decomposing such a form into its closable and singular part and using Bochner’s theorem it is possible to derive the central decomposition of the associated gauge-invariant quasi-free state on the boson $$C^*$$-Weyl algebra. The appearance of a classical field part of the boson system is studied in detail in the GNS- representation and shown to correspond to the so-called singular subspace of a natural enlargement of the one-boson test function space. In the example of Bose-Einstein condensation a non-trivial central decomposition (or equivalently a non-trivial classical field part) is directly related to the occurrence of the condensation phenomenon.

##### MSC:
 46N50 Applications of functional analysis in quantum physics 46L60 Applications of selfadjoint operator algebras to physics
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